*>I have a 1 ft square, 1/16 inch thick copper plate and am holding it in the *

*>middle of one edge. I lower it quickly into a rapidly boiling pan of *

*>water. How long will it be before it gets too hot for me to hold. I'd *

*>appreciate pointers as to how to calculate this time. TIA*

The problem you have is one of 'thermal diffusivity'. For a simple

one-dimensional case, the problem can be worked out with calculus. You need

to know two parameters of the copper plate, it's heat capacity per unit

volume in terms of Joules/cm^3 (don't confuse this with the heat capacity

per unit mass. If you have heat capacity per unit mass, multiply it by the

density of the substance to get heat capacity per unit volume), and the

second parameter is thermal conductivity (Watts/m-C).

The key here is that it isn't a 'steady-state' problem, but a transient one.

For a simple case, assume the copper at one edge is at 100C at time 0

seconds, and the entire rest of the plate is at 20C at the same time 0. So

heat begins to conduct from the hot edge toward material a short distance

away at the rate of....

Q = conductivity*area*(temperature gradient)

(remember the area is that area perpendicular to the heat flow, so it is the

'thin' edge of your plate, not the area of the plate as you look at it from

the side). The temperature gradient is simply the difference in temperature

divided by the distance between two points (dT/dX).

How fast that amount of heat raises the temperature of the cooler material

is a function

dT/dt = Q/ (area*dX*volumetric heat capacity)

Substituting for Q, we get...

dT/dt = (conductivity / volumetric heat capacity)*d^2T / dX^2

(sorry about this, but dT on the left is the change in temperature with

respect to time, while dT on the right is the difference in temperature

between two points along the plate, not the same thing, so don't try to

combine them with algebra). The term '(conductivity / volumetric heat

capacity)' is used often enough that it is given it's own name and often is

listed separately in tables for materials as 'thermal diffusivity'.

To find out how much the temperature will rise at some distance dX to rise

over a given period, takes a lot of calculus for even a 'simple, flat

plate'. Most calculations require you to convert the various values to

dimensionless numbers and a lot of higher math.

But, if you're handy with Excel or a similar spreadsheet, you can figure out

a very good approximation by breaking down the plate into a whole lot of

thin strips running perpendicular to the heat flow. Suppose you chose to

break it down into 10 strips. Set up the first column of the spreadsheet

with a series of numbers from 0 to some arbitrarily large number of seconds

(say, count from 0 to 500 downward). In the next column put the temperature

of the first strip at time zero (say, 100C). Skip a column for the moment

and in the fourth column put the temperature of the second strip at time

zero (say, 20C). Now, back to the third column put in a formula to

calculate the heat transfer between the two strips based on these two

temperatures (something like '=<conductivity>*<area>*(b1-d1)/<width> ').

Now, take the third and forth column and copy them to the fifth and sixth.

Repeat copying like this until you have a total of ten copies across.

Copy all but the first column down to the row right underneath it. But now

you have to *change* all the temperature columns except the first one. The

new temperature at time '1' for each strip is the previous temperature, plus

the temperature rise caused by the heat addition (for d2, something like

*'=d1+c1*<deltatime>/(<heat capacity>*<area>*<width>)' ).*
Now, take the second row and copy it down as many times as you want. You

should see the second strip's temperature rise first, followed by the other

strips as the heat travels across the strips in a sort of 'bow wave' affect.

To answer your original question, follow the temperature of the last 'strip'

and decide at what temperature it will be 'too hot to hold', and simply look

across to the left to see what time step this temperature is reached.

Sometimes this sort of 'explicit integration' can be much easier than all

the calculus that would normally be required. If you follow Nick's posts

often, you'll recognize this is a technique he often uses with VB to solve

transient problems. I find for 'one-of' types of calcs, it is easy to just

set up several columns in Excel and then copy/paste for as many iterations

(each row as a time-step) as needed. Of course, the finer the time step,

and the thinner the 'strips' are (although you'll need more columns to

simulate all the heat flows and temperatures in more steps), the more

accurate the answer.

A good text on transient heat flow is "Principles of Heat Transfer 3rd

Edition" by Kreith. Although a lot of it uses differential equations and

calculus, some of the simple cases such as flat plate can be solved with the

use of a couple of charts and converting the problem to 'dimensionless

numbers and ratios'.

daestrom

*> This is a complex problem, and I probably shouldn't have hastily posted*

*> the first-order solution I suggested before, since it could be wildly*

*> inaccurate. The general problem of transient heat conduction was*

*> studied by Fourier and he was inspired to develop Fourier analysis. At*

*> the time it was perhaps the most advanced technique for analyzing and*

*> solving this problem, and it turned out to have a lot of applications*

*> and is still widely used.*

Yes, the texts I have on the subject also discuss periodic variations in the

applied heat and calculating the temperature through the material T(t,x) as

a quite complicated function. But in some simple cases, such as starting

with a uniform temperature, and applying the heat as a step change from zero

to some value at t=0, it simplifies quite a bit.

*> Now with computers, numerical analysis is*

*> common. The iterative numerical method proposed by daestrom is a good*

*> technique. I've used it and it works.*

Yes, before the advant of computers for solving this sort of problem in

simple step-wise integration, quite a lot of the work was done in calculus

and Fourier analysis. And a few assumptions to eliminate higher order

effects was common. But hey, we got to the moon and built the atom-bomb

using slide-rules, so it worked.

The only 'trick' to using iterative numerical methods is knowing when to

throw away the answers ;-)

daestrom

Thanks to everyone. I think I have an adequate idea of the problem now.

I'm getting old and the brain ain't what it was. It looked simple and I was

bothered that I couldn't figure it out myself. It is now evident that it

isn't that simple so I don't feel so bad.

I do use Excel quite a lot and I will try the technique suggested by

Daestrom.

I want to be able to estimate how the temperature and pressure of a heated

and confined gas will vary with time when being heated (or cooled) by water

inside a heat exchanger finned pipe. Considering surface area,

temperatures, specific heat and gravity etc for the air/metal interface is

easy but adding the mass and volume of the metal in the heat exchanger slows

things down a lot. Adding the time it will take for the heat to travel from

the water to the ends of the fins makes it even slower. Tough old life is

it not? It does help keep the old brain functioning though.

Thanks again to all who replied. I'll stop bothering you, for a while at

least.

Alan C

*>>I have a 1 ft square, 1/16 inch thick copper plate and am holding it in *

*>>the*

*> The problem you have is one of 'thermal diffusivity'. For a simple *

*> one-dimensional case, the problem can be worked out with calculus. You *

*> need to know two parameters of the copper plate, it's heat capacity per *

*> unit volume in terms of Joules/cm^3 (don't confuse this with the heat *

*> capacity per unit mass. If you have heat capacity per unit mass, multiply *

*> it by the density of the substance to get heat capacity per unit volume), *

*> and the second parameter is thermal conductivity (Watts/m-C).*

*> The key here is that it isn't a 'steady-state' problem, but a transient *

*> one. For a simple case, assume the copper at one edge is at 100C at time 0 *

*> seconds, and the entire rest of the plate is at 20C at the same time 0. *

*> So heat begins to conduct from the hot edge toward material a short *

*> distance away at the rate of....*

*> Q = conductivity*area*(temperature gradient)*

*> (remember the area is that area perpendicular to the heat flow, so it is *

*> the 'thin' edge of your plate, not the area of the plate as you look at it *

*> from the side). The temperature gradient is simply the difference in *

*> temperature divided by the distance between two points (dT/dX).*

*> How fast that amount of heat raises the temperature of the cooler material *

*> is a function*

*> dT/dt = Q/ (area*dX*volumetric heat capacity)*

*> Substituting for Q, we get...*

*> dT/dt = (conductivity / volumetric heat capacity)*d^2T / dX^2*

*> (sorry about this, but dT on the left is the change in temperature with *

*> respect to time, while dT on the right is the difference in temperature *

*> between two points along the plate, not the same thing, so don't try to *

*> combine them with algebra). The term '(conductivity / volumetric heat *

*> capacity)' is used often enough that it is given it's own name and often *

*> is listed separately in tables for materials as 'thermal diffusivity'.*

*> To find out how much the temperature will rise at some distance dX to rise *

*> over a given period, takes a lot of calculus for even a 'simple, flat *

*> plate'. Most calculations require you to convert the various values to *

*> dimensionless numbers and a lot of higher math.*

*> But, if you're handy with Excel or a similar spreadsheet, you can figure *

*> out a very good approximation by breaking down the plate into a whole lot *

*> of thin strips running perpendicular to the heat flow. Suppose you chose *

*> to break it down into 10 strips. Set up the first column of the *

*> spreadsheet with a series of numbers from 0 to some arbitrarily large *

*> number of seconds (say, count from 0 to 500 downward). In the next column *

*> put the temperature of the first strip at time zero (say, 100C). Skip a *

*> column for the moment and in the fourth column put the temperature of the *

*> second strip at time zero (say, 20C). Now, back to the third column put *

*> in a formula to calculate the heat transfer between the two strips based *

*> on these two temperatures (something like *

*> '=<conductivity>*<area>*(b1-d1)/<width> '). Now, take the third and forth *

*> column and copy them to the fifth and sixth. Repeat copying like this *

*> until you have a total of ten copies across.*

*> Copy all but the first column down to the row right underneath it. But *

*> now you have to *change* all the temperature columns except the first one. *

*> The new temperature at time '1' for each strip is the previous *

*> temperature, plus the temperature rise caused by the heat addition (for *

*> d2, something like '=d1+c1*<deltatime>/(<heat *

*> capacity>*<area>*<width>)' ).*

*> Now, take the second row and copy it down as many times as you want. You *

*> should see the second strip's temperature rise first, followed by the *

*> other strips as the heat travels across the strips in a sort of 'bow wave' *

*> affect. To answer your original question, follow the temperature of the *

*> last 'strip' and decide at what temperature it will be 'too hot to hold', *

*> and simply look across to the left to see what time step this temperature *

*> is reached.*

*> Sometimes this sort of 'explicit integration' can be much easier than all *

*> the calculus that would normally be required. If you follow Nick's posts *

*> often, you'll recognize this is a technique he often uses with VB to solve *

*> transient problems. I find for 'one-of' types of calcs, it is easy to *

*> just set up several columns in Excel and then copy/paste for as many *

*> iterations (each row as a time-step) as needed. Of course, the finer the *

*> time step, and the thinner the 'strips' are (although you'll need more *

*> columns to simulate all the heat flows and temperatures in more steps), *

*> the more accurate the answer.*

*> A good text on transient heat flow is "Principles of Heat Transfer 3rd *

*> Edition" by Kreith. Although a lot of it uses differential equations and *

*> calculus, some of the simple cases such as flat plate can be solved with *

*> the use of a couple of charts and converting the problem to 'dimensionless *

*> numbers and ratios'.*

*> daestrom*

*> *

> I appreciate the help but am still having a problem. I don't understand